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# FZERO procedure

Gives the F function expectation under complete spatial randomness (M.A. Mugglestone, S.A. Harding, B.Y.Y. Lee, P.J. Diggle & B.S. Rowlingson).

### Option

`PRINT` = string token What to print (`summary`); default `summ`

### Parameters

`DENSITY` = scalars Densities to use i.e. numbers of points per unit area; no default – this parameter must be set Vectors of distances to use; no default – this parameter must be set Variates to receive the expected values of the F nearest-neighbour distribution function under CSR

### Description

The F nearest-neighbour distribution function relates to the distribution of distances from each of a set of sample points covering the region of interest to the nearest event of an observed spatial point pattern (see Diggle 1983). (The procedure `FHAT` can be used to obtain an estimate of F given an observed pattern and a set of sample points.) The term complete spatial randomness (CSR) is used to represent the hypothesis that the overall density of events in a spatial point pattern is constant throughout the study region, and that the events are distributed independently and uniformly.

Under CSR, and ignoring edge effects, the F nearest-neighbour distribution function is given by

F(s) = 1 – exp(-i × density × (s2)),

where density is the overall density of events per unit area. (The F nearest-neighbour distribution function for a clustered (regular) pattern will tend to be smaller (larger) than values calculated using the above expression, at least for small distances.) The procedure `FZERO` evaluates this expression for a given density (specified using the parameter `DENSITY`) and a vector of distances (specified using the parameter `S`). (The procedure `PTINTENSITY` may be used to obtain the density of events in an observed pattern prior to using `FZERO.`) The output of the procedure is a vector containing the expected values of F under CSR for each distance in `S`. The values of the F function can be saved using the parameter `FVALUES`.

Printed output is controlled using the `PRINT` option. The default setting of `summary` prints the distances at which the F function is estimated, and the estimates themselves under the headings `S` and `FVALUES`.

Another nearest-neighbour distribution function, the G nearest-neighbour distribution function, relates to the distribution of distances from each event of a spatial point pattern to the nearest other event in the pattern (see Diggle 1983). (The procedure `GHAT` can be used to obtain an estimate of G for an observed pattern.) Under CSR, the F and G nearest-neighbour distribution functions are identical. The output from the procedure `FZERO` can, therefore, be compared to estimates of the G nearest-neighbour distribution function. (The G nearest-neighbour distribution function for a clustered (regular) pattern will tend to be larger (smaller) than the values given by the above expression, at least for small distances.)

Option: `PRINT`.

Parameters: `DENSITY`, `S`, `FVALUES`.

### Method

The `CALCULATE` directive is used to evaluate the expression for the expected value of the F function using the density of events specified by the parameter `DENSITY` and the set of distances in `S`.

### Action with `RESTRICT`

If `S` is restricted, only the subset of values specified by the restriction will be included in the calculations.

### Reference

Diggle, P.J. (1983). Statistical Analysis of Spatial Point Patterns. Academic Press, London.

Procedures: `FHAT`, `GHAT`, `KHAT`, `KSTHAT`, `K12HAT`.

Commands for: Spatial statistics.

### Example

```CAPTION     'FZERO example'; STYLE=meta
VARIATE     pinex,piney
0.09 0.91   0.02 0.71   0.03 0.62   0.18 0.61   0.03 0.52
0.02 0.41   0.16 0.35   0.13 0.33   0.13 0.27   0.03 0.21
0.13 0.14   0.08 0.11   0.02 0.02   0.18 0.98   0.31 0.89
0.22 0.58   0.13 0.52   0.21 0.38   0.23 0.27   0.23 0.11
0.41 0.98   0.44 0.97   0.42 0.93   0.42 0.48   0.43 0.36
0.59 0.92   0.63 0.92   0.63 0.88   0.66 0.88   0.58 0.83
0.53 0.69   0.52 0.68   0.49 0.58   0.52 0.48   0.52 0.09
0.58 0.06   0.68 0.66   0.68 0.63   0.67 0.53   0.67 0.48
0.67 0.41   0.68 0.34   0.66 0.24   0.73 0.27   0.74 0.11
0.78 0.06   0.79 0.02   0.86 0.03   0.84 0.88   0.94 0.89
0.95 0.83   0.79 0.79   0.84 0.71   0.83 0.68   0.86 0.65
0.79 0.61   0.93 0.48   0.83 0.42   0.93 0.31   0.93 0.23
0.97 0.64   0.96 0.64   0.96 0.61   0.96 0.57   0.97 0.38  :
VARIATE     xpoly; VALUES=!(0,1,1,0)
&           ypoly; VALUES=!(0,0,1,1)
PTINTENSITY [PRINT=*] Y=piney; X=pinex; YPOLYGON=ypoly; XPOLYGON=xpoly;\
DENSITY=density
VARIATE     s; VALUES=!(0.01,0.02...0.1)
FZERO       DENSITY=density; S=s
```
Updated on March 7, 2019